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| {{Refimprove|date=December 2009}}
| | Hello. Let me introduce the author. Her name is Emilia Shroyer but it's not the most feminine title out there. What I love doing is taking part in baseball but I haven't made a dime with it. I am a meter reader. His family lives in South Dakota but his spouse wants them to move.<br><br>Stop by my page: [http://www.ubi-cation.com/ubication/node/6056 std testing at home] |
| [[File:PolymerBrush.jpg|thumb|100px|Sample polymer brush]]
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| '''Polymer brush''' is a layer of [[polymer]]s attached with one end to a surface.<ref name="Milner">{{cite journal | last1 = Milner | first1 = S. T. | title = Polymer Brushes | journal = Science | volume = 251 | issue = 4996 | pages = 905 | year = 1991 | pmid = 17847384 | doi = 10.1126/science.251.4996.905|bibcode = 1991Sci...251..905M }}</ref> The brushes may be either in a solvent state, when the dangling chains are submerged into a solvent, or in a melt state, when the dangling chains completely fill up the space available. Additionally, there is a separate class of polyelectrolyte brushes, when the polymer chains themselves carry an electrostatic charge.
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| The brushes are often characterized by the high density of grafted chains. The limited space then leads to a strong extension of the chains, and unusual properties of the system. Brushes can be used to stabilize [[colloid]]s, reduce friction between surfaces, and to provide lubrication in artificial [[joint]]s.<ref name="Halperin">{{cite journal | last1 = Halperin | first1 = A. | last2 = Tirrell | first2 = M. | last3 = Lodge | first3 = T. P. | title = Tethered chains in polymer microstructures | volume = 100/1 | pages = 31 | year = 1992 | doi = 10.1007/BFb0051635 }}</ref>
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| Polymer brushes have been modeled with Monte Carlo methods,<ref name="Laradji">{{cite journal | last1 = Laradji | first1 = Mohamed | last2 = Guo | first2 = Hong | last3 = Zuckermann | first3 = Martin | title = Off-lattice Monte Carlo simulation of polymer brushes in good solvents | journal = Physical Review E | volume = 49 | pages = 3199 | year = 1994 | doi = 10.1103/PhysRevE.49.3199|bibcode = 1994PhRvE..49.3199L }}</ref> [[Brownian dynamics]] simulations<ref name="Kaznessis">{{cite journal | last1 = Kaznessis | first1 = Yiannis N. | last2 = Hill | first2 = Davide A. | last3 = Maginn | first3 = Edward J. | title = Molecular Dynamics Simulations of Polar Polymer Brushes | journal = Macromolecules | volume = 31 | pages = 3116 | year = 1998 | doi = 10.1021/ma9714934 |bibcode = 1998MaMol..31.3116K }}</ref> and molecular theories <ref name="Szleifer">{{cite journal | last1 = Szleifer | first1 = I | last2 = Carignano | first2 = MA | title = Tethered Polymer Layers | journal = Adv. Chem. Phys. | volume = XCIV | pages = 165 | year = 1996 | ISBN = 978-0-471-19143-8 }}</ref>
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| == Structure of a polymer brush ==
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| [[File:Tethered polymer chain.svg|thumb|300px|Polymer molecule within a brush. The drawing shows the chain elongation decreasing from the attachment point and vanishing at free end. The "blobs", schematized as circles, represent the (local) length scale at which the statistics of the chain change from a 3D random walk (at smaller length scales) to a 2D in-plane random walk and a 1D normal directed walk (at larger length scales).]]
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| Polymer molecules within a brush are stretched away from the attachment surface as a result of the fact that they repel each other (steric repulsion or osmotic pressure). More precisely,<ref name="MWC88">{{cite journal | last1 = Milner | first1 = S. T | last2 = Witten | first2 = T. A | last3 = Cates | first3 = M. E | title = A Parabolic Density Profile for Grafted Polymers | journal = Europhysics Letters (EPL) | volume = 5 | pages = 413 | year = 1988 | doi = 10.1209/0295-5075/5/5/006|bibcode = 1988EL......5..413M }}</ref> they are more elongated near the attachment point and unstretched at the free end, as depicted on the drawing.
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| More preciseley, within the approximation derived by Milner, Witten, Cates,<ref name="MWC88" /> the average density of all monomers in a given chain is always the same up to a prefactor:
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| <math>\phi(z,\rho)=\frac{\partial n}{\partial z}</math>
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| <math>n(z,\rho)=\frac{2N}{\pi}\arcsin\left(\frac{z}{\rho}\right)</math>
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| where <math>\rho</math> is the altitude of the end monomer and <math>N</math> the number of monomers per chain.
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| The averaged density profile <math>\epsilon(\rho)</math> of the end monomers of all attached chains, convoluted with the above density profile for one chain, determines the density profile of the brush as a whole:
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| <math>\phi(z)=\int_z^\infty \frac{\partial n(z,\rho)}{\partial z}\,\epsilon(\rho)\,{\rm d}\rho</math>
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| A '''dry brush''' has a uniform monomer density up to some altitude <math>H</math>. One can show <ref name="MWC89">{{cite journal | last1 = Milner | first1 = S. T | last2 = Witten | first2 = T. A | last3 = Cates | first3 = M. E | journal = Macromolecules | volume = 22 | pages = 853–861 | year = 1989 |bibcode = 1989MaMol..22..853M |doi = 10.1021/ma00192a057 }}</ref> that the corresponding end monomer density profile is given by:
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| <math>\epsilon_{\rm dry}(\rho,H)=\frac{\rho/H}{Na\sqrt{1-\rho^2/H^2}}</math>
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| where <math>a</math> is the monomer size.
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| The above monomer density profile <math>n(z,\rho)</math> for one single chain minimizes the total elastic energy of the brush,
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| <math>U=\int_0^\infty\epsilon(\rho)\,{\rm d}\rho\,\int_0^N\,{\rm d}n\,\frac{kT}{2Na^2}\left(\frac{\partial z(n,\rho)}{\partial n}\right)^2</math>
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| regardless of the end monomer density profile <math>\epsilon(\rho)</math>, as shown in.<ref name="ZhuBor1991">{{cite journal | last1 = Zhulina | last2 = Borisov | journal = J. Colloid Interface Sci. | volume = 44 | pages = 507–520 | year = 1991 }}</ref><ref name="Gay1997">{{cite journal | last1 = Gay | first1 = C. | title = Wetting of a polymer brush by a chemically identical polymer melt | journal = Macromolecules | volume = 30 | pages = 5939–5943 | year = 1997 |bibcode = 1997MaMol..30.5939G |doi = 10.1021/ma970107f }}</ref>
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| == From a dry brush to any brush ==
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| As a consequence,<ref name="Gay1997" /> the structure of any brush can be derived from the brush density profile <math>\phi(z)</math>. Indeed, the free end distribution is simply a convolution of the density profile with the free end distribution of a dry brush:
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| <math>\epsilon(\rho)=\int_\rho^\infty -\frac{{\rm d}\phi(H)}{{\rm d}H}\epsilon_{\rm dry}(\rho,H)</math>.
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| Correspondingly, the brush elastic free energy is given by:
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| <math>\frac{F_{\rm el}}{kT}=\frac{\pi^2}{24N^2a^5}\int_0^\infty\left\{-z^3\frac{{\rm d}\phi(z)}{{\rm d}z}\right\}{\rm d}z</math>.
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| This method has been used to derive wetting properties of polymer melts on polymer brushes of the same species <ref name="Gay1997" /> and to understand fine interpenetration asymmetries between copolymer lamellae<ref name="Leibler99asym">{{cite journal | last1 = Leibler | first1 = L | last2 = Gay | first2 = C | last3 = Erukhimovich | first3 = I | title = Conditions for the existence of non-centrosymmetric copolymer lamellar systems | journal = Europhysics Letters (EPL) | volume = 46 | pages = 549–554 | year = 1999 }}</ref> that may yield very unusual non-centrosymmetric [[lamellar structure]]s.<ref name="Goldacker99">{{cite journal | last1 = Goldacker | first1 = T | last2 = Abetz | first2 = V | last3 = Stadler | first3 = R | last4 = Erukhimovich | first4 = I | last5 = Leibler | first5 = L | title = Non-centrosymmetric superlattices in block copolymer blends | journal = Nature | volume = 398 | pages = 137 | year = 1999 |bibcode = 1999Natur.398..137G |doi = 10.1038/18191 }}</ref>
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| == References ==
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| <references/>
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| ==Further reading==
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| {{Empty section|date=March 2013}}
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| ==See also==
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| * [[dendronized polymer]]
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| {{DEFAULTSORT:Polymer Brush}}
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| [[Category:Condensed matter physics]]
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| [[Category:Soft matter]]
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| [[Category:Polymer chemistry]]
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Hello. Let me introduce the author. Her name is Emilia Shroyer but it's not the most feminine title out there. What I love doing is taking part in baseball but I haven't made a dime with it. I am a meter reader. His family lives in South Dakota but his spouse wants them to move.
Stop by my page: std testing at home